On Representations of Semigroups Having

Hypercube-like Cayley Graphs

Cody Cassiday and G. Stacey Staples∗

Abstract

The n-dimensional hypercube, or n-cube, is the Cayley graph of theAbelian group Z2

n. A number of combinatorially-interesting groups andsemigroups arise from modified hypercubes. The inherent combinatorialproperties of these groups and semigroups make them useful in a num-ber of contexts, including coding theory, graph theory, stochastic pro-cesses, and even quantum mechanics. In this paper, particular groupsand semigroups whose Cayley graphs are generalizations of hypercubes aredescribed, and their irreducible representations are characterized. Con-structions of faithful representations are also presented for each semigroup.The associated semigroup algebras are realized within the context of Clif-ford algebras.

AMS Subj. Class. 05E15, 15A66, 47L99Keywords: semigroups, combinatorics, hypercubes, representation theory,Clifford algebras

1 Introduction

Hypercubes are regular polytopes frequently arising in computer science andcombinatorics. They are intricately connected to Gray codes, and their graphsarise as Hasse diagrams of finite boolean algebras. Hamilton cycles in hyper-cubes correspond to cyclic Gray codes and have received significant attentionin light of the “Middle Level Conjecture” originally proposed by I. Havel [23].

On the classical stochastic side, random walks on hypercubes are useful inmodeling tree-structured parallel computations [8]. On the quantum side, hy-percubes often appear in quantum random walks, e.g. [1, 10]. Random walkson Clifford algebras have also been studied as random walks on directed hyper-cubes [15].

By considering specific generalizations of hypercubes, combinatorial prop-erties can be obtained for tackling a variety of problems in graph theory andcombinatorics [17, 21, 22]. By defining combinatorial raising and lowering oper-ators on the associated semigroup algebras, an operator calculus (OC) on graphs

∗Department of Mathematics and Statistics, Southern Illinois University Edwardsville,Edwardsville, IL 62026-1653, USA

1

is obtained, making graph-theoretic problems accessible to the tools of algebraic(quantum) probability [14, 16, 18, 19].

The goal of the current work is to describe some particular groups and semi-groups whose Cayley graphs are generalizations of hypercubes, to characterizetheir irreducible complex representations, and to provide constructions for faith-ful representations.

2 Signed, Directed, & Looped Hypercubes

Hypercubes play an important role throughout the operator calculus approach.The n-dimensional cube, or hypercube Qn, is the graph whose vertices are inone-to-one correspondence with the n-tuples of zeros and ones and whose edgesare the pairs of n-tuples that differ in exactly one position. This graph hasnatural applications in computer science, symbolic dynamics, and coding theory.The structure of the hypercube allows one to construct a random walk on thehypercube by “flipping” a randomly selected digit from 0 to 1 or vice versa.

Given two binary strings a = (a1a2 · · · an) and b = (b1b2 · · · bn), the Ham-ming distance between a and b, denoted dH(a, b), is defined as the number ofpositions at which the strings differ. That is,

dH(a, b) = |{i : 1 ≤ i ≤ n, ai 6= bi}|.

Let b be a block, or word, of length n; that is, let b be a sequence of n zerosand ones. The Hamming weight of b, denoted wH(b), is defined as the number ofones in the sequence. The binary sum of two such words is the sequence resultingfrom addition modulo-two of the two sequences. The Hamming distance betweentwo binary words is defined as the weight of their binary sum.

With Hamming distance defined, the formal definition of the hypercube Qncan be given.

Definition 2.1. The n-dimensional hypercube Qn is the graph whose verticesare the 2n n-tuples from {0, 1} and whose edges are defined by the rule

{v1, v2} ∈ E(Qn)⇔ wH(v1 ⊕ v2) = 1.

Here v1 ⊕ v2 is bitwise addition modulo-two, and wH is the Hamming weight.In other words, two vertices of the hypercube are adjacent if and only if theirHamming distance is 1.

Fixing the set B = {e1, . . . , en}, the power set of B is in one-to-one corre-spondence with the vertices of Qn via the binary subset representation

(a1a2 · · · an)↔ eI ⇔ ai =

{1 i ∈ I,0 otherwise.

Of particular interest are some variations on the traditional hypercube de-fined above. First, the looped hypercube is the pseudograph obtained from thetraditional hypercube Qn by appending a loop at each vertex. In particular,Qn◦ = (V◦, E◦), where V = V (Qn) and E = E(Qn) ∪ {(v, v) : v ∈ V (Qn)}.

2

Figure 2.1: Four-dimensional hypercube.

Definition 2.2. Let V denote the vertex set of the n-dimensional hypercube,Qn. Let αV denote the set obtained from V by appending the symbol α to eachvertex in V . The Hamming weight of α is taken to be zero. A signed hypercubeis a (possibly directed) graph, G, on vertex set V ∪ αV such that

(u,w) ∈ E(G)⇒ wH(u⊕ w) = 1.

With various notions of generalized hypercubes in mind, a few relevant ex-amples of finitely-generated semigroups can be given.

Let S04 denote the Abelian group generated by commutative generators{γ1, γ2, γ3, γ4} along with γ∅ satisfying γi

2 = γ∅ for each i. The Cayley graphof S04 is readily seen to be the four-dimensional hypercube of Figure 2.1.

Let J4 denote the Abelian group generated by γ∅ along with commutativegenerators {γ1, . . . , γ4} satisfying γi

2 = γi for each i ∈ {1, 2, 3, 4}. The Cayleygraph of J4 is then readily seen to be the four-dimensional looped hypercubeobtained by appending a loop to each vertex of the graph seen in Figure 2.1.

Let B30 denote the non-Abelian group generated by {γ1, γ2, γ3} along with γ∅and γα satisfying γi

2 = γα for each i, and γiγj = γαγjγi for i 6= j. The signedthree-dimensional hypercube of Figure 2.2 is the undirected graph underlyingthe Cayley graph of B30.

Hypercube generalizations appearing in this paper are summarized in Table2.

2.1 The blade group Bqp

Let B = {e1, . . . , en}, and let p and q be nonnegative integers such that p+ q =n. Let Bqp be the multiplicative group generated by B along with the elements

3

Figure 2.2: Three-dimensional signed hypercube.

(Semi) Generator GeneratorGroup Commutation SquaresBqp γiγj = γαγjγi {γ∅, . . . , γ∅︸ ︷︷ ︸

p

, γα, . . . , γα︸ ︷︷ ︸q

}

Sqp Abelian {γ∅, . . . , γ∅︸ ︷︷ ︸p

, γα, . . . , γα︸ ︷︷ ︸q

}

Gn γiγj = γαγjγi γi2 = 0γ , i = 1, . . . , n

Zn Abelian γi2 = 0γ , i = 1, . . . , n

Jn Abelian γi2 = γi, i = 1, . . . , n

Table 1: Semigroups summarized by generators.

(Semi) Directed? Signed? LoopedGroupBqp Yes Yes NoSqp No Only if q > 0. NoGn Yes No NoZn No No NoJn No No Yes

Table 2: Properties of hypercubes underlying semigroups discussed.

{e∅, eα}, subject to the following generating relations: for all x ∈ B ∪ {e∅, eα},e∅ x = x e∅ = x,

eα x = x eα,

e∅2 = eα

2 = e∅,

4

and

eiej =

eα ejei if i 6= j,

e∅ if i = j ≤ p,eα if p+ 1 ≤ i = j ≤ n.

The group Bqp is referred to as the blade group 1 of signature (p, q).

Let 2[n] denote the power set of the n-set, [n] = {1, 2, . . . , n}, used as indicesof the generators in B. Elements of 2[n] are assumed to be canonically orderedby

I ≺ J ⇔∑i∈I

2i−1 <∑j∈J

2j−1. (2.1)

Note that the ordering is inherited from the binary subset representation ofintegers.

Remark 2.3. The order of the blade group Bqp is 2p+q+1 as seen by noting the

form of its elements, i.e., Bqp = {eI , eα eI : I ∈ 2[n]}.To simplify multiplication within Bqp, some additional mappings will be use-

ful. For fixed positive integer j, define the map µj : 2[n] → N0 by

µj(I) = |{i ∈ I : i > j}|. (2.2)

In other words, µj(I) is the counting measure of the set {i ∈ I : i > j}.

Definition 2.4. The product signature map ϑ : 2[n]× 2[n] → {e∅, eα} is definedby

ϑ(I, J) = eα(µp(I∩J)+

∑j∈J µj(I)). (2.3)

Applying multi-index notation to the generators B according to the orderedproduct

eI =∏i∈I

ei (2.4)

for arbitrary I ∈ 2[n], the multiplicative group Bqp is now seen to be deter-

mined by the multi-indexed set {eI , eαeI : I ∈ 2[n]} along with the associativemultiplication defined by

eI eJ = ϑ(I, J)eI4J , (2.5)

where I4J = (I ∪ J) \ (I ∩ J) denotes set-symmetric difference. Inverses in Bqpare given by

eI−1 = ϑ(I, I)eI

sinceeIϑ(I, I)eI = ϑ(I, I)2eI4I = e∅.

1The elements of this group are analogous to the “basis blades” of a Clifford (Grassmann)algebra.

5

Elements of the form eI are called positive, while elements of the form eαeIare called negative. Positive elements of Bqp are now canonically ordered by

eI ≺ eJ ⇔ I ≺ J

using the ordering on 2[n] given by (2.1).An element eI ∈ Bqp is said to be even if |I| = 2k for some nonnegative

integer k. Otherwise, eI is said to be odd.

Lemma 2.5. The collection of even elements of Bqp forms a normal subgroup,

denoted Bqp+.

Bqp+ C Bqp.

Proof. First, note that multiplicative identity, e∅ is indexed by a set of sizezero so that Bqp

+ contains the identity. Secondly, the inverse of any element eIis indexed by the same subset so that Bqp

+ is closed with respect to inverses.Finally, the symmetric difference of two sets of even cardinality is also of evencardinality so that Bqp

+ is closed under multiplication. Thus, Bqp+ is a subgroup

of Bqp.To see that Bqp

+ is a normal subgroup, let eI ∈ Bqp be fixed and consider con-

jugation of elements of Bqp+. That is, consider eIBqp

+eI−1. Choosing arbitrary

eJ ∈ Bqp+, one finds

eI eJ eI−1 = ϑ(I, I)eI eJ eI = ϑ(I, I)eIϑ(J, I)eJ4I

= ϑ(I, I)ϑ(J, I)ϑ(I, J4I)eI4(J4I)

= ϑ(I, I)ϑ(J, I)ϑ(I, J4I)eJ ∈ Bqp+.

Hence, the result.

Allowing commutativity of generators leads to another combinatorially in-teresting group referred to as the “Abelian blade group.”

Definition 2.6. The Abelian blade group, Sqp , is defined as the abelian group oforder 2n+1 generated by the collection S = {ςi : 1 ≤ i ≤ n} along with elements{ς∅, ςα} satisfying the following generating relations: for all x ∈ S ∪ {ς∅, ςα},

ς∅ x = x ς∅ = x,

ςα x = x ςα,

ς∅2 = ςα

2 = ς∅,

and

ςiςj =

ςjςi if 1 ≤ i 6= j ≤ n,ς∅ if 1 ≤ i = j ≤ p,ςα if p+ 1 ≤ i = j ≤ n.

The quotient group algebra RSqp/〈ςα + ς∅〉 is canonically isomorphic to thesymmetric-Clifford algebra C`p,qsym appearing in [21], where it is used to inducehomogeneous random walks on hypercubes.

6

3 Group Representations

All group and semigroup representations considered in this paper are complex.A representation of a given group, G, is a homomorphism ρ : G → GLn(C).The degree of this representation is n, and the representation space is the spaceCn on which the elements of GLn(C) act.

Given a representation ρ and a subspace W of Cn, we say W is G-invariantif ρ(g)W ⊆ W for every g ∈ G. If the only invariant spaces are {0} and Cn,the representation is said to be irreducible. The character of a representation,χ : G→ C, is defined by χ(g) = tr(ρ(g)).

A fundamental result in group representation theory [20] is that a represen-tation ρ with character χ is irreducible if and only if χ satisfies

(χ|χ) =1

|G|∑g∈G

χ(g)χ(g) = 1.

Two representations ρ and r of a group G are said to be isomorphic if thereexists an invertible mapping f : Cn → Cn such that

f ◦ ρ = r ◦ f.

Lemma 3.1. The group Bqp has at least 2p+q distinct degree-1 irreducible rep-resentations.

Proof. First, any representation ρ of degree 1 must satisfy ρ(e∅) = 1. We claimthat in the case p + q > 1, this implies ρ(eα) = ρ(e∅) = 1. To see this, notethat eα

2 = e∅ clearly implies ρ(eα) = ±1. Suppose ρ(eα) = −1, and considerthe following cases:

1. The case p = 0 or q = 0. In this case, either ei2 = e∅ for all 1 ≤ i ≤ p+ q

or ei2 = eα for all 1 ≤ i ≤ p + q. In either case, since p + q > 1, one

considers the product ei ej for 1 ≤ i 6= j ≤ p + q. By anticommutativity,(ei ej)

2 = eα, but ρ(eα) = −1 guarantees a contradiction.

2. The case p, q ≥ 1. In this case, one considers a pair, i, j, satisfying 1 ≤i ≤ p and p + 1 ≤ j ≤ p + q. Then ei

2 = e∅, ej2 = eα, and (ei ej)

2 = e∅,again leading to a contradiction.

Let J ∈ 2[p+q] denote a multi-index. A degree-1 representation ρJ is definedby setting ρJ(e∅) = ρJ(eα) = 1, and for 1 ≤ i ≤ p+ q, setting

ρJ(ei) =

{1 i ∈ J,−1 i /∈ J

By considering the number of distinct subsets J , it follows immediately thatthe total number of representations created this way is 2p+q. These representa-tions are clearly irreducible and distinct, i.e., pairwise non-isomorphic.

7

Remark 3.2. The groups B10 and B01 are instances of Abelian blade groups con-sidered in Section 3.1. Irreducible representations of B10 are characterized inExample 3.4.

Recall that given a group G, the conjugacy class of an element g ∈ G is theset

Cl(g) = {hgh−1 : h ∈ G}.

A well-known result in representation theory says that the number of irreduciblerepresentations of a group G is equal to the number of its conjugacy classes [20].This provides a useful tool for establishing the following result.

Theorem 3.3. Given the group Bqp the number of conjugacy classes and subse-quently the number of irreducible representations is given by the formula

κ = 2p+q + 1 + c,

where c = p+ q (mod 2).

Proof. If p+ q = 1 the formula trivially works.Suppose p + q 6= 1 and denote the center of Bqp by Z(Bqp). If g ∈ Bqp\Z(Bqp)

then it is easily seen that

Cl(g) = {hgh−1 : h ∈ Bqp}

= {eIge−1I : I ∈ 2[p+q]}.

Combining the following facts: eI−1 ∈ {eI , eαeI}, eIg ∈ {geI , eαgeI}, and

eI2 ∈ {eα, e∅}, one sees that Cl(g) = {g, eαg}.

Further, if g ∈ Z(Bqp), then eαg ∈ Z(Bqp) and Cl(g) = {g}. Hence,

κ =|Bqp\Z(Bqp)|

2+ |Z(Bqp)|.

Therefore to finish the proof we need to understand the order of the center.Let eI ∈ Bqp be arbitrary. If I = ∅ one can see that e∅ ∈ Z(Bqp).

Now suppose I 6= ∅. Let I = {i1, . . . , ih} for h even. Then,

eiheI = (eα)h−1eIeih= eαeIeih .

Whence, eI 6∈ Z(Bqp).Assume now that I = {i1, ...ih} 6= {1, 2, ..., p+ q} for h odd. Then there is a

natural number ` such that ` /∈ I and

e`eI = eαeIe`.

Thus, eI /∈ Z(Bqp).Finally, suppose I = {1, 2, ..., p + q} = [p + q] for p + q odd. It is claimed

thateJe[p+q] = e[p+q]eJ

8

for every indexing set J ⊆ I. To see this, note that if J = ∅ the result triviallyholds. By way of induction on the cardinality of J , assume J = {j}, setµ+ = |{i ∈ [p+ q] : i < j}|, and set µ− = |{i ∈ [p+ q] : i > j}|. Then,

e[p+q]ej = (eα)µ−+µ+

eje[p+q]

= (eα)p+q−1eje[p+q]

= eje[p+q].

Suppose e[p+q]eJ = eJe[p+q] for some multi-index cardinality |J |. The tasknow is to show e[p+q]eJeh = eJehe[p+q] for some natural number h 6∈ J. Fromthe inductive hypothesis we know

(e[p+q]eJ)eh = (eJe[p+q])eh

and from the basis step, we know e[p+q]eh = ehe[p+q].Combining these two facts and using associativity of the group operation,

e[p+q]eJeh = eJe[p+q]eh = eJehe[p+q].

Hence, by induction, eJe[p+q] = e[p+q]eJ for all J ∈ 2[p+q].If p+ q is even, then Z(Bqp) = {eα, e∅}. In this case,

κ =|Bqp\Z(Bqp)|

2+ |Z(Bqp)|

= (2p+q − 1) + 2

= 2p+q + 1.

If p+ q is odd, then Z(Bqp) = {eα, e∅, e[p+q], eαe[p+q]}, which gives

κ =|Bqp\Z(Bqp)|

2+ |Z(Bqp)|

= (2p+q − 2) + 4

= 2p+q + 2.

Example 3.4. Consider the group B10. In this case, two non-faithful irreduciblerepresentations are constructed as in the proof of Lemma 3.1. Two more faith-ful irreducible representations are found in agreement with Theorem 3.3. Thefour representations are listed in the left table of Figure 3.1. Four irreduciblerepresentations of B01 are similarly constructed in the right table of Figure 3.1.No faithful irreducible representations exist in this case. It is not difficult toverify that the representations are distinct for each group.

The above example could have been completed using real representations,although none would be faithful. When p+ q > 1, Bqp is non-Abelian, and hencehas no faithful degree-1 representation regardless of representation space.

Another well known result in group representation theory is the following,found in [20].

9

B10 e∅ eα e1 eα e1ρ∅ 1 1 1 1ρ{1} 1 1 −1 −1δ1 1 −1 ı −ıδ2 1 −1 −ı ı

B01 e∅ eα e1 eα e1ρ∅ 1 1 1 1ρ{1} 1 1 −1 −1δ1 1 −1 1 −1δ2 1 −1 −1 1

Figure 3.1: Irreducible degree-1 representations of B10 (left) and B01 (right).

Lemma 3.5. Let G be a finite group having κ irreducible representations. Foreach i = 1, . . . , κ, let ni denote the degree of the ith irreducible representationof G. Then,

|G| =κ∑i=1

ni2.

Given the group Bqp there are always 2p+q distinct irreducible representa-tions of degree 1. The remaining irreducible (complex) representations are nowenumerated in the next theorem.

Theorem 3.6. If p + q = 2k > 1, then Bqp has one irreducible representation

of degree 2k. If p + q = 2k + 1, then Bqp has two irreducible representations of

degree 2k. Moreover, all of these irreducible representations are faithful exceptwhen p is odd and q is even.

Proof. This will be treated in two cases. First is the case of p + q even. Sincep+q is even, one can write p+q = 2k for some k ∈ N. Define τ : Bqp → GL2k(C)by

τ(ej) =

σx⊗(j−1) ⊗ σz ⊗ σ0⊗(k−j) 1 ≤ j ≤ k, j ≤ p

ı(σx⊗(j−1) ⊗ σz ⊗ σ0⊗(k−j)

)1 ≤ j ≤ k, j > p

σx⊗(j−k−1) ⊗ σy ⊗ σ0⊗(2k−j) k + 1 ≤ j ≤ 2k, j ≤ p

ı(σx⊗(j−k−1) ⊗ σy ⊗ σ0⊗(2k−j)

)k + 1 ≤ j ≤ 2k, j > p.

(3.1)

Setting τ(e∅) = σ0⊗k and τ(eα) = −σ0⊗k, this extends by multiplication to all

of Bqp. More specifically, for any multi-index I, τ(eI) =∏`∈I

τ(e`), and τ(eαeI) =

−τ(eI).This representation is clearly well defined. To verify that this representation

is irreducible, let ξ be the character of τ . Let eI ∈ Bqp be arbitrary. Lettingσ(`) ∈ {σ0, σx, σy, σz} for each ` = 1, . . . , k, one writes

ξ(eI) = tr(τ(eI))

= tr

(u

k⊗`=1

σ(`)

)= u

∏`∈I

tr(σ(`))

10

for some unit u ∈ {±1,±ı}. Since tr(σx) = tr(σy) = tr(σz) = 0, and tr(σ0) = 2,it follows that ξ(eI) = ξ(eαeI) = 0 unless I = ∅, in which case ξ(e∅) = 2k andξ(eae∅) = −2k. Now,

(ξ|ξ) =1

|Bqp|∑g∈Bqp

ξ(g)ξ(g)

=1

22k+1((ξ(e∅))

2 + (ξ(eαe∅))2)

=1

22k+1(2)(22k) = 1.

Thus, τ is irreducible.To see that the representation (3.1) is faithful, consider the kernel:

ker(τ) = {eI : τ(eI) = σ0⊗k}.

Noting that τ(Bqp) is a subgroup of GL2k(C), we begin by showing that the

center of this subgroup consists only of elements having the form ±uσ0⊗k, whereu ∈ {±1,±ı}. To begin, the center of τ(Bqp) is

Z(τ(Bqp)) = {τ(eE) : τ(eE)τ(eJ) = τ(eJ)τ(eE), ∀J ∈ 2[p+q]}.

Suppose an element of Z(τ(Bqp)) is of the form M = uσ(1)⊗ . . .⊗σ(k), wherefor some index h, σ(h) 6= σ0 but σ(h+1) = · · · = σ(k) = σ0. If σ(h) = σz or σythen we can see

τ(e{h,h+k}) = u(σ0⊗(h−1) ⊗ σx ⊗ σ0(k−h)),

which will anti-commute with M .If σ(h) = σx, an element anti-commuting with M is given by

τ(e{1,k+1,2,k+2,...,h−1,k+h−1,h}) =h−1∏j=1

(−ı)(σ0⊗(j−1) ⊗ σx ⊗ σ0⊗(k−j))

(σx⊗(h−1) ⊗ σz ⊗ σ0⊗(k−h))= u

(σx⊗(h−1) ⊗ σ0⊗(k−h+1)

)(σx⊗(h−1) ⊗ σz ⊗ σ0⊗(k−h)

)= uσ0

⊗(h−1) ⊗ σz ⊗ σ0⊗(k−h).

This proves that every element of Z(τ(Bqp)) is of the form uσ0⊗k for u ∈

{±1,±i}.By construction, τ(e∅) and τ(eα) are elements of Z(τ(Bqp)). Suppose E is a

non-empty indexing set, and to the contrary suppose τ(eE) ∈ Z(τ(Bqp)), so that

τ(eE) = uσ0⊗k. It is not difficult to see that eE 6∈ Z(Bqp), so there exists an

integer m such thateEem = eαemeE .

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Applying τ reveals

τ(eEem) = uσ0⊗kτ(em) 6= −u τ(em)σ0

⊗k = τ(eαemeE).

This contradicts the homomorphism property of τ . We conclude then thateE 6∈ Z(τ(Bqp)), which means

Z(τ(Bqp)) = {τ(e∅), τ(eα)}.

However, only one of these, τ(e∅), is σ0⊗k. Thus, ker(τ) = {e∅}. Since the

kernel is trivial, τ is faithful.Now suppose p+ q = 2k + 1 is odd; more specifically, suppose p is even and

q is odd. Let τ : Bqp → GL2k(C) be defined by

τ(ej) =

σx⊗(j−1) ⊗ σz ⊗ σ0⊗(k−j) 1 ≤ j ≤ k, j ≤ p

ı(σx⊗(j−1) ⊗ σz ⊗ σ0⊗(k−j)

)1 ≤ j ≤ k, j > p

σx⊗(j−k−1) ⊗ σy ⊗ σ0⊗(2k−j) k + 1 ≤ j ≤ 2k, j ≤ p

ı(σx⊗(j−k−1) ⊗ σy ⊗ σ0⊗(2k−j)

)k + 1 ≤ j ≤ 2k, j > p

σx⊗k j = 2k + 1, j ≤ p

ı(σx⊗k) j = 2k + 1, j > p.

(3.2)

Setting τ(e∅) = σ0⊗k and τ(eαei) = −τ(ei), this extends by multiplication to

all of Bqp.To check irreducibility of τ , let ξ denote the character of τ . It is already

known that ξ(eI) = 0 in all but the extreme case I = ∅. Further, since e[p+q] is

in the center of the group, its image under τ must be of the for uσ0⊗k for some

scalar u ∈ {±1,±ı}. It thereby follows that

(ξ|ξ) =1

|Bqp|∑g∈Bqp

ξ(g)ξ(g)

=1

22k+2

((ξ(e∅))

2 + (ξ(eα))2)

+1

22k+2

(ξ(e[2k+1])ξ(e[2k+1]) + ξ(eαe[2k+1])ξ(eαe[2k+1])

)=

1

(22k+2)

(22k + 22k + 22k + 22k

)= 1.

Hence, τ is irreducible.Recall that when p + q is odd, Z(Bqp) = {eα, e∅, e[p+q], eαe[p+q]}. In light

of the proof that τ was faithful for p + q even, showing that e[p+q] /∈ ker(τ) issufficient to show that τ is faithful. Computing e[p+q], one finds

e[p+q] 7→ ıq(σzσyσx)⊗k = ıq(σ0⊗k), (3.3)

12

so that

τ(e[p+q]) =

{ı(σ0

⊗k) if q ≡ 1 (mod 4),

−ı(σ0⊗k) if q ≡ 3 (mod 4).(3.4)

It follows that ker(τ) is trivial.Finally, in the case p is odd and q is even, the construction of (3.2) is again

used. This representation is again irreducible, and τ(e[p+q]) = ıq(σ0⊗k), as seen

in Equation 3.3. In this case, however, one has

τ(e[p+q]) =

{σ0⊗k = τ(e∅) when q ≡ 0 (mod 4),

−σ0⊗k = τ(eα e∅) when q ≡ 2 (mod 4),(3.5)

so that the representation is not faithful.Recalling that the order of Bqp is equal to the sum of the squares of degrees

of irreducible representations, there remains one irreducible representation ofBqp in the case p+ q is odd: the complex conjugate of τ . This representation isgiven explicitly by

τ(ej) =

σx⊗(j−1) ⊗ σz ⊗ σ0⊗(k−j) 1 ≤ j ≤ k, j ≤ p

−ı(σx⊗(j−1) ⊗ σz ⊗ σ0⊗(k−j)

)1 ≤ j ≤ k, j > p

−σx⊗(j−k−1) ⊗ σy ⊗ σ0⊗(2k−j) k + 1 ≤ j ≤ 2k, j ≤ pı(σx⊗(j−k−1) ⊗ σy ⊗ σ0⊗(2k−j)

)k + 1 ≤ j ≤ 2k, j > p

σx⊗k j = 2k + 1, j ≤ p

−ı(σx⊗k) j = 2k + 1, j > p,

where, τ(e∅) = σ0⊗k and τ(eαei) = −τ(ei). This extends by multiplication to

all of Bqp.To see that τ is not isomorphic to τ , one considers the action of τ on e[p+q].

In particular, (3.4) and (3.5) imply

τ(e[p+q]) =

σ0⊗k = τ(e[p+q]) when q ≡ 0 (mod 4),

−ı(σ0⊗k) = −τ(e[p+q]) when q ≡ 1 (mod 4),

−σ0⊗k = τ(e[p+q]) when q ≡ 2 (mod 4),

ı(σ0⊗k) = −τ(e[p+q]) when q ≡ 3 (mod 4).

Suppose there exists an invertible linear transformation f ∈ GL(C2k) satisfyingf ◦ τ = τ ◦ f . Then, the cases q ≡ 1 (mod 4) and q ≡ 3 (mod 4) imply

f ◦(ı(σ0⊗k)) = −ı

(σ0⊗k) ◦ f ⇒ f(v) = −v, ∀v ∈ C2k ,

which contradicts f ◦ τ(e∅) = σ0⊗k. Similarly, in the cases q ≡ 0 (mod 4) and

q ≡ 2 (mod 4),

f ◦(σ0⊗k) =

(σ0⊗k) ◦ f ⇒ f(v) = v, ∀v ∈ C2k ,

contradicting f ◦ τ = τ ◦ f , since τ 6= τ .

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It becomes evident in the case p ≡ 1 (mod 2) and q ≡ 0 (mod 2) thatin order to obtain a faithful representation of Bqp, one must pass to a largerrepresentation space. It is not difficult to show that a faithful representation isgiven by defining τ : Bqp → GL2k+1(C) by

τ(ej) =

σx⊗(j−1) ⊗ σz ⊗ σ0⊗(k−j+1) 1 ≤ j ≤ k, j ≤ p

ı(σx⊗(j−1) ⊗ σz ⊗ σ0⊗(k−j+1)

)1 ≤ j ≤ k, j > p

σx⊗(j−k−1) ⊗ σy ⊗ σ0⊗(2k−j) k + 1 ≤ j ≤ 2k, j ≤ p

ı(σx⊗(j−k−1) ⊗ σy ⊗ σ0⊗(2k−j)

)k + 1 ≤ j ≤ 2k, j > p

σx⊗(k+1) j = 2k + 1, j ≤ p

ı(σx⊗(k+1)

)j = 2k + 1, j > p.

Setting τ(e∅) = σ0⊗(k+1) and τ(eαei) = −τ(ei), this is again extended by

multiplication to all of Bqp.

3.1 The Abelian blade group Sqp

In the case of the Abelian blade group, Sqp , commutativity removes any hopeof finding an irreducible faithful representation except for the case of S10 ∼=B10, as noted in Example 3.4. The order of Sqp is 2p+q+1, and its irreduciblerepresentations are found as follows.

As in Lemma 3.1, let J ∈ 2[p+q] denote a multi-index. A degree-1 repre-sentation ρJ is defined by setting ρJ(ς∅) = ρJ(ςα) = 1, and for 1 ≤ i ≤ p + q,setting

ρJ(ςi) =

{1 i ∈ J,−1 i /∈ J

Similarly, a degree-1 representation, δJ , is obtained for each multi index J bysetting δJ(ς∅) = 1, δJ(ςα) = −1, and

δJ(ς`) =

1 1 ≤ ` ≤ p and ` ∈ J−1 1 ≤ ` ≤ p and ` /∈ Jı p+ 1 ≤ ` ≤ p+ q and ` ∈ J,−ı p+ 1 ≤ ` ≤ p+ q and ` /∈ J.

Hence, all 2p+q+1 degree-1 irreducible representations are obtained.One can find a faithful representation of order 2p+q. Let ϕ be given by

multiplicative extension of

ϕ(ςj) = uσ⊗(j−1)0 ⊗ σz ⊗ σ⊗(p+q−j)0 ,

where u = 1 or u = ı depending on j. This representation is clearly faithfulby construction. A meaningful question to ask is whether a smaller faithful

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representation exists. This question is answered in the affirmative by definingthe degree-2(p+ q) faithful representation, r : Sqp → GL2(p+q)(C) as follows:

r(ςI) = u

A1 0 · · · 0

0 A2

......

. . . 00 · · · 0 An

.

Here, each Aj is a 2× 2 matrix given by

Aj =

{σx if j ∈ I,σ0 otherwise,

and u is a complex unit determined by

u =

{1 if ςI

2 = ς∅,

ı if ςI2 = ςα.

4 Semigroup Representations

Essential definitions and notational conventions for semigroup representationtheory follow the formalism of Izhakian, Rhodes, and Steinberg [7]. As previ-ously noted, all semigroup representation spaces are complex.

Given a semigroup S, two elements a, b ∈ S are said to be J-equivalent(written a J b) if SaS = SbS. The set of all things J-equivalent to a ∈ S forms aJ-class. The J-classes partition the semigroup S. A J-class is said to be regularif it contains an idempotent.

For every idempotent e of a semigroup S, we call Ge the maximal subgroupof S at e where Ge = {invertible elements of eSe}. Two idempotent elementse, f ∈ S are said to be isomorphic if there exists an x ∈ eSf and x∗ ∈ fSe suchthat xx∗ = e and x∗x = f .

The regular J-classes will play a large role in determining the number ofirreducible representations of a semigroup S. Before we give the exact numberto expect, we need a few more results. The following useful lemma can be foundin [4].

Lemma 4.1. If e, f ∈ S are isomorphic idempotents, then Ge ' Gf . Moreover,e and f are isomorphic if and only if e J f .

For a semigroup S we define a representation to be a homomorphism to theset of endomorphisms of Cn, which can be realized as n×n matrices with entriesin C. In other words, a representation ρ of S is a homomorphism

ρ : S → End(Cn).

The familiar terminology of a faithful representation, trivial representationand character of a representation follows. The idea of an irreducible semigroup

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representation is again the same, except we require that the representation isnot constantly 0.

The next theorem, based on results of Clifford-Suchkewitch [4] and Munn (asfound in Rhodes and Zalcstein [13]), will be useful for determining the numberof irreducible representations.

Theorem 4.2. Let G1, ..., Gm be a choice of exactly one maximal subgroupfrom each regular J-class of S. Then, letting ki denote the number of conjugacy

classes of Gi, the number of irreducible representations of S ism∑i=1

ki.

4.1 Null blade semigroups Gn and Zn

By modifying the multiplication in Bqp such that generators square to zero,one obtains a non-Abelian semigroup generated by null squares. The principaldifference from this point forward is a lack of multiplicative inverses for elementsin the algebraic structures.

Definition 4.3. Let Gn denote the null blade semigroup defined as the semi-group generated by the collection G = {γi : 1 ≤ i ≤ n} along with {γ∅, γα, 0γ}satisfying the following generating relations: for all x ∈ G ∪ {γ∅, γα, 0γ},

γ∅ x = x γ∅ = x,

γα x = x γα,

0γ x = x 0γ = 0γ ,

γ∅2 = γα

2 = γ∅,

and

γiγj =

{0γ if and only if i = j,

γα γjγi i 6= j.

Define the antisymmetric product signature map φ : 2[n]×2[n] → {γ∅, γα} by

φ(I, J) = γα∑j∈J µj(I).

Remark 4.4. Note that the product signature map defined by (2.3) can beextended to G × G and written in terms of φ as

ϑ(I, J) = γαµp(I∩J)+φ(I,J).

Hence, ϑ has a decomposition into signature-dependent and signature-independentparts.

Applying multi-index notation to the generators G = {γi : 1 ≤ i ≤ n}according to the ordered product

γI =∏i∈I

γi

16

for arbitrary I ∈ 2[n], the multiplicative semigroup Gn is now seen to be de-termined by the multi-indexed set {0γ} ∪ {γαγI , γI : I ∈ 2[n]} along with theassociative multiplication defined by

γI γJ =

{γα

∑i∈I µi(J)γI∪J I ∩ J = ∅,

0 otherwise.

Note that the order of the null blade semigroup is |Gn| = 2n+1 + 1. Thenext combinatorial algebra can now be defined.

Definition 4.5. For fixed positive integer n, the null blade algebra is defined asthe real semigroup algebra RGn/ 〈0γ , γα + γ∅〉, denoted B`∧n for convenience.

It now becomes clear that the null blade algebra B`∧n is canonically isomor-phic to the Grassmann (exterior) algebra

∧Rn.

Theorem 4.6. For any natural number n, there are three irreducible represen-tations of Gn.

Proof. First, we will classify every J-class of Gn, then identify which are regular.From there we will compute the maximal subgroups of a choice of distinct idem-potents. Then, using the formula above we will find the number of irreduciblerepresentations.

Let γI ∈ Gn be such that γI 6= 0γ but otherwise arbitrary. Then,

GnγIGn = {s1 γI s2 : s1, s2 ∈ Gn}= {γE , γαγE , 0γ : I ⊆ E}.

Similarly,Gn(0γ)Gn = {0γ}.

It follows that for every w ∈ Gn, the set of all things J -equivalent to wis simply {w, γαw}. The number of J-classes is thus 2n + 1. However we areonly concerned with the regular J-classes. The only idempotent elements ofGn are 0γ and γ∅. Thus, the regular J-classes are {γ∅, γα} and {0γ}. The twomaximal subgroups are

Gγ∅ = {invertible elements of γ∅Gnγ∅} = {γ∅, γα},

andG0γ = {invertible elemenets of 0γGn0γ} = {0γ}.

The trivial group, G0γ , has one conjugacy class, while Gγ∅ is an Abeliangroup of order 2, consequently having two conjugacy classes. Thus, the numberof irreducible representations of Gn is three.

Definition 4.7. Let Zn denote the Abelian null blade semigroup defined as thesemigroup generated by the collection C = {ζi : 1 ≤ i ≤ n} along with {ζ∅, 0ζ}

17

satisfying the following generating relations: for all x ∈ C ∪ {ζ∅, 0ζ},

ζ∅ x = x ζ∅ = x,

0ζ x = x 0ζ = 0ζ ,

ζ∅2 = 0ζ ,

and

ζiζj =

{0ζ if and only if i = j,

ζjζi i 6= j.

The Abelian null blade semigroup is of particular interest, as its associatedsemigroup algebra is canonically isomorphic to the zeon algebra. Properties ofthis algebra have been considered and applied in a number of works in recentyears, including [5, 6, 14, 16, 17, 18, 22].

Using nearly the same proof as above, it becomes apparent that Zn has twocopies of the trivial group as maximal subgroups, and thus has two irreduciblerepresentations, regardless of n.

The irreducible representations of both Zn and Gn are almost immediatelyobvious. For arbitrary n, define degree-1 representations θ, ρ0, and ρ1 of Gn by

θ(s) = 1, ∀s ∈ Gn.

ρ0(s) =

1 s = γ∅,−1 s = γα,0 otherwise.

ρ1(s) =

1 s = γ∅,1 s = γα,0 otherwise.

Note that these representations are clearly not faithful.In Zn, the irreducible representations are simply θ and the degree-1 repre-

sentation ρ given by

ρ(s) =

{1 s = ζ∅,0 otherwise.

Given an arbitrary natural number n, there is a faithful representation τ ofGn of order 2n given by

τ(γi) = σx⊗(i−1) ⊗ η ⊗ σ0⊗(n−i),

where η = σz + ıσy =

(1 1−1 −1

).

Similarly, a faithful representation ψ of Zn is given by

ψ(ζi) = σ0⊗(i−1) ⊗ η ⊗ σ0⊗(n−i).

4.2 The idempotent blade semigroup Jn

In the idempotent blade semigroup, generators are idempotent. The resultingsemigroup algebra is isomorphic to the “idem-Clifford” algebra C`nidem usedto define column-idempotent adjacency matrices of graphs [18]. These oper-ators can be used to symbolically represent collections of cycles as productsof algebraic elements. In such a product, a graph’s edges are associated withidempotents to avoid “double counting” in enumeration problems.

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Definition 4.8. Let Jn denote the Abelian semigroup of order 2n generatedby the collection E = {εi : 1 ≤ i ≤ n} along with ε∅ satisfying the followinggenerating relations: for all x ∈ E ∪ {ε∅} and for all i, j ∈ {1, . . . , n},

ε∅ x = x ε∅ = x,

εi2 = εi, and

εiεj = εjεi.

Theorem 4.9. For any natural number n, there are 2n irreducible representa-tions of Jn.

Proof. The proof method follows the same format as the previous one. EveryJ-class of Jn is classified, and then the regular classes are identified. From eachregular J-class one idempotent element is chosen and the maximal subgroup ate is computed.

Each element is in its own J-class with no equivalent idempotent elements,giving |Jn| = 2n unique idempotents. The maximal subgroups are found to beGε∅ = {ε∅} and GεI = {εI} for arbitrary non-trivial idempotent εI .

Enumerating the idempotent elements {f1, ..., f2n} and letting ki be thenumber of conjugacy classes in Gfi , the number of irreducible representationsis thus

2n∑i=1

ki =

2n∑i=1

1 = 2n.

It would be nice if we were able to supply faithful representations of Jn,even if they are reducible. This isn’t too difficult, let τ : Jn → End(Cn+1) bedefined on the set {εi} by

τ(εi) = (aijk),

where

aijk =

{1 j = k 6= i,

0 otherwise.(4.1)

In other words, (aijk) is the matrix with ones on the diagonal except in the ith

position, and zeros elsewhere. These matrices are all idempotent and commutepairwise. This is extended by multiplication to all of Jn so that

τ(εI) = (aIjk),

where

aIjk =

{1 j = k /∈ I,0 otherwise.

Remark 4.10. For each i = 1, . . . , n, the matrix defined in (4.1) representsa hyperplane projection in Cn+1. In particular, the ith matrix represents aprojection onto the hyperplane orthogonal to the ith unit coordinate vector ofCn+1 .

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5 Combinatorial Graded Semigroup Algebras

Beginning with a finite multiplicative semigroup, S, the semigroup algebra of Sover R is the algebra RS whose additive group is the Abelian group of formalR-linear combinations of elements of S, i.e.,

RS =

{∑s∈S

αss : αs ∈ R

}

and whose multiplication operation is defined by linear extension of the groupmultiplication operation of S. This definition restricts in a natural way to groupalgebras.

Given a (complex) representation ρ of a finite semigroup S, let ρ denote therepresentation of the |S|-dimensional semigroup algebra CS given by

x =∑s∈G

αss⇒ ρ(x) =∑s∈S

αsρ(s),

where αs ∈ C for each s ∈ S.It is known that ρ is irreducible if and only if ρ is irreducible [13], so that

the irreducible representations of S are in one-to-one correspondence with theirreducible representations of CS. In particular, if ρ is an irreducible represen-tation of CS, an irreducible representation of S is obtained by restricting ρ tothe elements of S.

Classifying the irreducible representations of Bqp and Sqp thereby classifiesthe irreducible representations of the group algebras RBqp and RSqp . Similarly,classifying the irreducible representations for Gn,Zn and Jn classifies the irre-ducible representations of the semigroup algebras RGn, RZn, and RJn. Takingquotients reveals the algebras introduced in column 4 of Table 3.

To summarize:

• The Clifford algebra C`p,q (p + q > 1) is canonically isomorphic to theblade group quotient algebra RBqp/〈eα + e∅〉. Considering the degree-1

representations, ρJ(e∅) = ρJ(eα) = 1 for all J ∈ 2[p+q]. It then be-comes clear that passing to the quotient has no effect on the number ofirreducible representations. On the other hand, the higher-dimensionalirreducible representations satisfy τ(e∅ + eα) = 0 a priori, so that rep-resentations of the group algebra are precisely the representations of thequotient algebra 2

2While results are stated here within the context of complex representation spaces, par-ticular representations are, in fact, real. For example, the construction given in (3.1) forBqp when p = q yields elements of GL2k (R). Degrees of faithful representations then vary bygroup signature. A detailed treatment of smallest fields for representation spaces and minimaldegrees of faithful representations is outside the scope of this work, as the goal is to enumerateirreducible complex representations for combinatorial semigroups. Such details for the quo-tient group algebra RBqp/〈eα + e∅〉 are covered by known results on matrix representations ofClifford algebras (e.g., Bott periodicity) [2, 3, 9, 11, 12].

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• The symmetric Clifford algebra, C`p,qsym [19, 21], is canonically isomorphicto the Abelian blade group algebra RSqp/〈ςα + ς∅〉. By similar reasoningto that for the blade group quotient algebra, the number of irreduciblerepresentations is unchanged by considering the quotient.

• The Grassmann exterior algebra,∧

Rn, is canonically isomorphic to thenull blade semigroup algebra B`∧n = RGn/ 〈0γ , γα + γ∅〉. This algebra isisomorphic to the algebra of fermion creation (or annihilation) operators.

• The n-particle zeon algebra [5, 6, 16, 22] is canonically isomorphic to theAbelian null blade semigroup algebra RZn/ 〈0ζ〉. This algebra is isomor-phic to an algebra of commuting lowering or raising (annihilation or cre-ation) operators.

• The idem-Clifford algebra, C`nidem [17, 19], is canonically isomorphic tothe idempotent-generated semigroup algebra RJn.

Example 5.1. Regarding γ∅ and γα as 1 and −1, respectively, the signedhypercube seen in Figure 2.2 is the undirected graph underlying C`0,3. Similarly,Figure 2.1 underlies the symmetric Clifford algebra C`4,0sym.

Group or Quotient IsomorphicSemigroup Algebra Algebra AlgebraBqp RBqp RBqp/〈eα + e∅〉 C`p,qSqp RSqp RSqp/〈ςα + ς∅〉 C`p,qsymGn RGn RGn/ 〈0γ , γα + γ∅〉

∧Rn

Zn RZn RZn/ 〈0ζ〉 C`nnil

Jn RJn RJn C`nidem

Table 3: Semigroup algebras.

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